Equivalence Theorem for Convergence of Convolution Powers of a Probability Measure on Locally Compact Topological Groups
by
Liu Jin'e
Department of Statistics, Shandong Institute of Economics, Jinan 250014, China
Coauthors: Zhang Hui

Equivalence Theorem for Convergence of
Convolution Powers of a Probability Measure
on Locally Compact Topological Groups

Liu Jin'e
(Department of Statistics, Shandong Institute of Economics,
Jinan 250014, China)
Zhang Hui
(Department of Mathematics, Qufu Normal University,
Qufu 273165, Shandong, China)

Our main result is:

Theorem   Let S be a locally compact group and \mu in P(S), denote \Lambda(\mu) be the set of all limit points of {\muⁿ}. Let \lambda = \lambda² be the unit element of \Lambda(\mu) and F be the minimal closed subgroup containing S_\mu . The following conditions are equivalent:

(i)  {\muⁿ} is weakly convergent;

(ii)  the set limS_\muⁿ is not empty;

(iii)  limS_\muⁿ=[`lim]S_\muⁿ;

(iv)  F=[[`( \cup _{n >= 1}(S_\muⁿ)(S_\muⁿ)^-1)]];

(v)  S_\mu is not contained in any proper coset of any closed normal subgroup of F ;

(vi)  S_\mu is not contained in any proper coset of S_\lambda in F ;

(vii)   for allx in F, B in B(S), \lambda(Bx^-1)=\lambda(x^-1B) = \lambda(B) and S_\lambda=F .

To prove this theorem, we need following key lemmas:

Lemma 1    Let S be a locally compact semigroup and \mu in P(S) . {\muⁿ} is tight. Denote \Lambda(\mu) be the set of all limit points of {\muⁿ} . Then

(i)  \Lambda(\mu) is a subgroup of P(S);

(ii)   for allk >= 1, there exist \mu_k in \Lambda(\mu) , such that \mu^k*\mu_k=\mu_k*\mu^k=\lambda , where \lambda is the unit element of \Lambda(\mu) ;

(iii)  Denote Q(\mu)[( /\ ) || =]{\muⁿ: n=1, 2, 3, ... }, then Q(\mu)*\lambda subset \Lambda(\mu) .

Lemma 2    If the idempotent in S₁ which is a completely simple semigroup is unique. Then S₁ is a group.

Lemma 3    Let S be a locally compact semigroup and N be a completely simple semigroup of measures on S. Then

È \mu in N  S\mu   = È \mu in N  S\mu

and they are both completely simple semigroups.

Lemma 4    Let T be a closed subgroup of S for t in T. If H is a closed subset of T, then H·t and t·H are both closed.

Lemma 5    Let S be a locally compact group and \mu in P(S) . Denote

Q(\mu) /\ =   {\mun: n=1, 2, ... },

\Lambda(\mu) /\ =   {\nu:\nu is a limit point of {\mun}},

G /\ =   supp\Lambda(\mu) /\ =   È \nu in \Lambda(\mu)  S\nu    ,

\Lambda is the identity of \Lambda(\mu) . Then

(i)  G is a closed subgroup of S and G= \cup _{\nu in \Lambda(\mu)}S_\nu ;

(ii)  S_\lambda is a normal subgroup of G;

(iii)   for all\nu in \Lambda(\mu) , g in S_v , \nu = \lambda* \delta_g = \delta_g * \lambda , where \delta_g is the point mass at g .

Lemma 6    Let \mu in P(S) , S a locally compact group. Then

(i)  G=[`(lim_n)] S_\muⁿ=F , where F is the minimal closed group containg S_\mu ;

(ii)  If limS_\muⁿ =/= \emptyset , then G=limS_\muⁿ .

Date received: June 15, 1998